The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(\varepsilon ^{-1} \lg n)$ suffices. This is almost matched by a lower bound of $s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d \geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d \ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.

Bird's eye view (BEV) is widely adopted by most of the current point cloud detectors due to the applicability of well-explored 2D detection techniques. However, existing methods obtain BEV features by simply collapsing voxel or point features along the height dimension, which causes the heavy loss of 3D spatial information. To alleviate the information loss, we propose a novel point cloud detection network based on a Multi-level feature dimensionality reduction strategy, called MDRNet. In MDRNet, the Spatial-aware Dimensionality Reduction (SDR) is designed to dynamically focus on the valuable parts of the object during voxel-to-BEV feature transformation. Furthermore, the Multi-level Spatial Residuals (MSR) is proposed to fuse the multi-level spatial information in the BEV feature maps. Extensive experiments on nuScenes show that the proposed method outperforms the state-of-the-art methods. The code will be available upon publication.

The most effective dimensionality reduction procedures produce interpretable features from the raw input space while also providing good performance for downstream supervised learning tasks. For many methods, this requires optimizing one or more hyperparameters for a specific task, which can limit generalizability. In this study we propose sparse compressed agglomeration (SparCA), a novel dimensionality reduction procedure that involves a multistep hierarchical feature grouping, compression, and feature selection process. We demonstrate the characteristics and performance of the SparCA method across heterogenous synthetic and real-world datasets, including images, natural language, and single cell gene expression data. Our results show that SparCA is applicable to a wide range of data types, produces highly interpretable features, and shows compelling performance on downstream supervised learning tasks without the need for hyperparameter tuning.

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With the recent surge in big data analytics for hyper-dimensional data there is a renewed interest in dimensionality reduction techniques for machine learning applications. In order for these methods to improve performance gains and understanding of the underlying data, a proper metric needs to be identified. This step is often overlooked and metrics are typically chosen without consideration of the underlying geometry of the data. In this paper, we present a method for incorporating elastic metrics into the t-distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP). We apply our method to functional data, which is uniquely characterized by rotations, parameterization, and scale. If these properties are ignored, they can lead to incorrect analysis and poor classification performance. Through our method we demonstrate improved performance on shape identification tasks for three benchmark data sets (MPEG-7, Car data set, and Plane data set of Thankoor), where we achieve 0.77, 0.95, and 1.00 F1 score, respectively.

Deep learning using neural networks is an effective technique for generating models of complex data. However, training such models can be expensive when networks have large model capacity resulting from a large number of layers and nodes. For training in such a computationally prohibitive regime, dimensionality reduction techniques ease the computational burden, and allow implementations of more robust networks. We propose a novel type of such dimensionality reduction via a new deep learning architecture based on fast matrix multiplication of a Kronecker product decomposition; in particular our network construction can be viewed as a Kronecker product-induced sparsification of an "extended" fully connected network. Analysis and practical examples show that this architecture allows a neural network to be trained and implemented with a significant reduction in computational time and resources, while achieving a similar error level compared to a traditional feedforward neural network.

Become a Data Scientist expert! Everything you need to get the job you want! "Dimensionality Reduction: Machine Learning with Python" is likely a guide or tutorial that focuses on the topic of dimensionality reduction in the context of machine learning. In machine learning, dimensionality reduction is the process of reducing the number of features in a dataset while preserving as much of the important information as possible. This is often necessary because high-dimensional datasets can be difficult to work with, and can lead to problems such as overfitting and increased computational complexity. This guide likely covers these techniques with some implementation of these techniques using python libraries like numpy, scikit-learn and matplotlib .

Abstract: The weighted Euclidean distance between two vectors is a Euclidean distance where the contribution of each dimension is scaled by a given non-negative weight. The Johnson-Lindenstrauss (JL) lemma can be easily adapted to the weighted Euclidean distance if weights are known at construction time. Given a set of n vectors with dimension d, it suffices to scale each dimension of the input vectors according to the weights, and then apply any standard JL reduction: the weighted Euclidean distance between pairs of vectors is preserved within a multiplicative factor ε with high probability. However, this is not the case when weights are provided after the dimensionality reduction. In this paper, we show that by applying a linear map from real vectors to a complex vector space, it is possible to update the compressed vectors so that the weighted Euclidean distances between pairs of points can be computed within a multiplicative factor ε, even when weights are provided after the dimensionality reduction.

High-dimensional spatio-temporal dynamics can often be encoded in a low-dimensional subspace. Engineering applications for modeling, characterization, design, and control of such large-scale systems often rely on dimensionality reduction to make solutions computationally tractable in real-time. Common existing paradigms for dimensionality reduction include linear methods, such as the singular value decomposition (SVD), and nonlinear methods, such as variants of convolutional autoencoders (CAE). However, these encoding techniques lack the ability to efficiently represent the complexity associated with spatio-temporal data, which often requires variable geometry, non-uniform grid resolution, adaptive meshing, and/or parametric dependencies. To resolve these practical engineering challenges, we propose a general framework called Neural Implicit Flow (NIF) that enables a mesh-agnostic, low-rank representation of large-scale, parametric, spatial-temporal data. NIF consists of two modified multilayer perceptrons (MLPs): (i) ShapeNet, which isolates and represents the spatial complexity, and (ii) ParameterNet, which accounts for any other input complexity, including parametric dependencies, time, and sensor measurements. We demonstrate the utility of NIF for parametric surrogate modeling, enabling the interpretable representation and compression of complex spatio-temporal dynamics, efficient many-spatial-query tasks, and improved generalization performance for sparse reconstruction.

Classical methods for acoustic scene mapping require the estimation of time difference of arrival (TDOA) between microphones. Unfortunately, TDOA estimation is very sensitive to reverberation and additive noise. We introduce an unsupervised data-driven approach that exploits the natural structure of the data. Our method builds upon local conformal autoencoders (LOCA) - an offline deep learning scheme for learning standardized data coordinates from measurements. Our experimental setup includes a microphone array that measures the transmitted sound source at multiple locations across the acoustic enclosure. We demonstrate that LOCA learns a representation that is isometric to the spatial locations of the microphones. The performance of our method is evaluated using a series of realistic simulations and compared with other dimensionality-reduction schemes. We further assess the influence of reverberation on the results of LOCA and show that it demonstrates considerable robustness.